One day last week I was chatting to a friend of mine who is a casual fan of the show and trying to explain, with my sketchy knowledge, about the early years and why Ian and Barbara didn't just plot a course for home after being 'kidnapped'. I might be wrong, but my answer was that something was broken on the TARDIS meaning that other than 'take off' and 'land' the Doctor had no control over destination and just had to hope for the best. This is right, right?

Anyway, it got me thinking about how I think this would be a far more interesting idea to use now than just having them decide to visit certain times and locations and would force the writers to be a lot more inventive with their stories. The Doctor couldn't be whizzing around answering text messages on the psychic paper and the departures of the companions could be a lot more interesting, as could their initial location. It would mean a lot more variance than the huge amounts of Earth based stories we get. 92 episodes have aired since the show returned, 67 (my own rough count) of those have been at least partly or entirely set on Earth and that isn't including ones set in space around earth (Series 1, I'm looking at you!). That's a huge amount. Taking that option away from them would mean a whole lot more variance in the locations, and they have shown in the past that they can do 'alien' locations without blowing the budget. It would also be really interesting as a viewer to not have any idea where they might end up in the next story. This would also make the tendency to lean on timey wimey arcs harder to do as well.

I think the downside for the viewer may be whether or not it would now be believable, though. The Doctor seems to be able to do anything these days and what he can't fix for himself, his sonic can instead. It might be a bit hard for viewers to swallow the idea that something broke on the TARDIS and he can't just put it right. Though if it works for the chameleon circuit...

I think an element of that would be good. I agree that it might be difficult to sustain to the degree it was at the start, but there could be something equivalent to the Fourth Doctor having to use the randomiser. Perhaps something that he can override, but there are good reasons not to most of the time.

What they haven't done much in NuWho are really good planets - rewatching The Keys of Marinus reinforced how effective that sort of thing could be. They've had New Earth, which is pretty much just a riff on Old Earth, the Oodsphere and a few others, but they're usually human-related. I'd love to see NuWho, with all its resources, tackle a totally alien planet and its inhabitants.

The explanation for why the Doctor couldn't take Ian and Barbara home varied a bit in the early days. At first, it was supposed to be that they'd left in too much of a hurry, and the Doctor needed to calculate exactly where they were before he could pinpoint a destination. At one point, Susan says "It's not that Grandfather can't steer the ship, but he does get so forgetful." Eventually, they seemed to settle on the guidance system being broken.

Of course, we know now that the journeys weren't quite as random as they seemed at the time

I completely agree that there needs to be a little more "chaos" in the operation of the TARDIS. One of the key things of the (early) years was the fact that the Doctor couldn't navigate terribly well. I also think we need more "alien". Keeping coming back to Earth might be a sensible budget decision but it seriously limits the options (as Terrence D1cks pointed out) to alien invasion, mad scientist or plague. I think some more planets (For example; Planet of the Ood) gives a much better dynamic.

They definitely need to explore more Space. There has been too much Earth over the years and I think you start to lose the fact that the TARDIS can go anywhere at all. I'd also like it to be more random. All the landings seem far too thought out. I liked when they had no idea where they would land and then had to go explore.

I liked the fact that the Doctor never knew when or we he would end up but I think a question that was hard to answer every time was why they stuck around when they could have just up and left. That isn't something the Doctor does much of these days as he is always looking for the adventure in everything so it wouldn't be as much of an issue now I don't suppose, but I think going somewhere with purpose means that that never really has to be dealt with now. I'm grand either way, I guess, though I think not knowing is always more exciting.

That was addressed pretty successfully in most of the very early stories, though it became a bit of a stretch later on. Then again, in some stories (eg The Romans) they hung around for a peaceful holiday, and events overtook them.

I like the idea of their landings being random. I think it makes it far more interesting, and as said in the OP, takes the option for big huge storyarcs (which I'm not a fan of) off the table. There is that thing in the early days where they had to find a reason for the Doctor to stay where he was rather than just leaving, but I think with the First Doctor especially, that was his nature. He wanted away from the troubles and I think Barbara and Ian are the main reason he started taking more of a caring interest. They forced him to because it was in their human nature to do so themselves. These days the Doctor can't leave anything alone and so they wouldn't have that problem. As soon as he finds out there is something to stick his nose into he is off and at it. Also, as Nyki points out, they need to have more planets and non Earth locations. Having random materialisations might cut that out a bit. Or at least give us more historicals.